Hypothetically speaking, if an incoming Attorney General wished to stop an ongoing investigation without cancelling it and tried to make the investigation grind to a halt by removing funding, would it be possible for the House to fund the investigation if they wished to see it completed?

This hypothetical situation so much more complicated than necessary. If you are referring to the Mueller investigation, the AG can simply end it, just like the chief of police can tell his officers to stop looking into a crime at any time, there is no reason for any maneuvering. The house can do any investigation it wants without tagging the AG office, it just has to subpoena anything and anyone it wants without any probable cause. They also do not need to maneuver.
– Frank CedenoNov 8 at 15:46

@FrankCedeno an interesting question would be whether the House could use the incomplete results of the special counsel investigation and to pick up where it left off, as it were, or even hire the same lawyer to manage its investigation. I do not know whether that would be possible, impossible, or contingent on some successful maneuvering.
– phoogNov 8 at 16:44

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@phoog, it is easy to think that things go on in congress or the office of the president like it is for us lowly people. It does not, congress can absolutely do anything they want like medieval kings. The restraint that keeps them in check is the fury of the people at the next election and fellow congresspersons. They can subpoena the report and question anyone about it. They can ignore the power of any of the other branches legally. I say legally because someone will bring up examples of on branch saying they have authority over the other one, however, you can also find counter examples
– Frank CedenoNov 8 at 18:34

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@FrankCedeno - Actually, he can't "just end it" - he has to justify his reasons for doing so. The regulations and authorizations that Mueller is operating under stipulate that he can only be fired with cause. That's part of the "Independent" designation for this position. Having a mere pretext as "cause" would almost certainly expose Trump even more to obstruction of justice charges.
– PoloHoleSetNov 8 at 22:46

@PoloHoleSet, I know you will never listen. The consequences of firing mueller are very serious for Trump, but they are not criminal, they are political. The Senate can use the phrase "Obstruction" in an impeachment but not as a violation of a law or regulation. They will not point to any statute but will say it because they can say anything is a high crime. So if Trump can get away with it politically he will fire Mueller, but no one is going to show up in his office with handcuffs.
– Frank CedenoNov 9 at 12:16

1 Answer
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The House of Representatives has its own budget. It determines how that budget is spent, although the budget itself has to be approved in a budget bill with Senate approval and either presidential acquiescence or a veto override. Within that budget, it could fund an investigation.

That's not really necessary though. The House also has the ability, even the duty, of providing oversight. It can investigate the president directly. And unlike a special counsel, its investigation would be mostly public.

In the specific current example of the Mueller investigation, cutting funding without ending the investigation just leaves the problem open. I suspect that it is more likely that they will tell Mueller to issue a report that can be published. The reason is that so far Mueller does not seem to have any evidence on the president that he is supposedly investigating. They want that to be published. They will then use that against any future House investigation. They will call the investigation a witch hunt after Mueller's investigation cleared the president. They will do so even if Mueller disagrees with that characterization. This is speculative, but entirely consistent with their previous behavior and statements.

Another step that they can take now that Jeff Sessions is no longer in the way is to start investigating Democratic politicians and leaking information about them. Sessions wouldn't sign off on the "Lock her up" investigation of Hillary Clinton, but the acting Attorney General probably will.